1. Field of the Invention
The present invention relates to a method and an apparatus for machining a non-circular workpiece, and more particularly to a method and an apparatus for machining a non-circular workpiece wherein a Fourier series is used in calculation of command values which are used for controlling a cutting tool of the apparatus.
2. Discussion of the Prior Art
A machining apparatus of the above-mentioned type is disclosed in the U.S. patent application No. 07/471,392, now U.S. Pat. No. 5,054,340 which was assigned to the applicant of this application.
In the apparatus, profile data defining a desired final shape of a non-circular workpiece is expanded into a Fourier series to obtain low frequency components of the profile data for actuating a low speed actuator such as a linear motor and high frequency components of the profile data for actuating a high speed actuator such as a piezoelectric actuator. A cutting tool of the apparatus is moved by the low speed actuator and the high speed actuator simultaneously. As described above, Fourier transformation is carried out to separate the profile data into low and high frequency components for driving the linear motor and the piezoelectric actuator, respectively. The Fourier transformation can also be used to improve the operational characteristic of the servo control system. The low frequency components used for moving the linear motor are referred to as low frequency command values hereinafter while the high frequency components used for moving the piezoelectric actuator are referred to as high frequency command values.
Namely, the apparatus is provided with a tool mount table lineally moved by a linear motor, and a cutting tool is mounted on the tool mount table through a piezoelectric actuator to be moved by the piezoelectric actuator with respect to the tool mount table. The cutting tool is moved by the linear motor and the piezoelectric actuator while a non-circular workpiece is rotated at a predetermined speed. The linear motor is driven in accordance with low frequency command values while the piezoelectric actuator is driven in accordance with high frequency command values. To obtain the low frequency command values and the high frequency command values, the profile data f(.theta.) is expanded into a Fourier series in a numerical controller of the apparatus in such a manner as given by the following expression: ##EQU1##
The Fourier coefficients a.sub.0, a.sub.1, b.sub.1. . . b.sub.6 of the above expression are found out by Fourier transformation. In the above example, expansion is made to sixth degree.
Then, components for zero to second degrees are assigned to the linear motor while components for third to sixth degrees are assigned to the piezoelectric actuator. Accordingly, low frequency command values XTL(.theta.) for driving the linear motor and high frequency command values XTP(.theta.) for driving the piezoelectric actuator are respectively calculated using following equations: ##EQU2##
The linear motor is driven in accordance with the low frequency command values XTL(.theta.) while the piezoelectric actuator is driven in accordance with the high frequency command values XTP(.theta.). As a result, the non-circular workpiece is machined into a desired profile.
In such apparatus, a desired shape of a non-circular workpiece is determined based upon a blue print thereof and is expressed by plural sets of data each of which includes an angular position (.theta..sub.n), a desired radius (r.sub.n) of the workpiece thereat, and an allowable error (.DELTA.r.sub.n) with respect to the desired radius, as shown in FIG. 7. The data is prepared for each of plural angular positions which have a predetermined angular interval of 5.degree. to 10.degree..
A Fourier series defines the desired profile are then calculated using the profile data. Since the expansion of the Fourier series is carried out within a limited number, for example to sixth degree, the profile defined by the Fourier series inevitably contains an error .epsilon., as shown in FIG. 1 (a). Although the error can be reduced by increasing the number of expansion, a problem arises that the acceleration component of command values calculated using the Fourier series is increased. The fluctuation of the acceleration component is also increased. When the acceleration component and the fluctuation thereof are large, the follow delay of the tool becomes large, whereby the machining accuracy is deteriorated.
Further, in cases where an allowable error changes in accordance with the rotational angle of the workpiece, as shown in FIG. 1 (b), the number of expansion has to be increased so that the error becomes smaller than the minimum value Emin of the allowable error .DELTA.r. Therefore, the number of the expansion cannot be reduced in such case.